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The Clay Mathematics Institute officially designated the title Millennium Problem for the seven unsolved mathematical problems, the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem, Riemann hypothesis, Yang–Mills existence and mass gap, and the Poincaré conjecture at the ...
First, it can be false in practice. A theoretical polynomial algorithm may have extremely large constant factors or exponents, rendering it impractical. For example, the problem of deciding whether a graph G contains H as a minor, where H is fixed, can be solved in a running time of O(n 2), [25] where n is the number of vertices in G.
The others are experimental, meaning that there is a difficulty in creating an experiment to test a proposed theory or investigate a phenomenon in greater detail. There are still some questions beyond the Standard Model of physics , such as the strong CP problem , neutrino mass , matter–antimatter asymmetry , and the nature of dark matter and ...
A polynomial-time problem can be very difficult to solve in practice if the polynomial's degree or constants are large enough. In addition, information-theoretic security provides cryptographic methods that cannot be broken even with unlimited computing power. "A large-scale quantum computer would be able to efficiently solve NP-complete problems."
BPP is one of the largest practical classes of problems, meaning most problems of interest in BPP have efficient probabilistic algorithms that can be run quickly on real modern machines. BPP also contains P , the class of problems solvable in polynomial time with a deterministic machine, since a deterministic machine is a special case of a ...
In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) is the problem of determining if there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the variables of a given Boolean formula ...
There are several proposed solutions to solve the strong CP problem. The most well-known is Peccei–Quinn theory , involving new scalar particles called axions . A newer, more radical approach not requiring the axion is a theory involving two time dimensions first proposed in 1998 by Bars, Deliduman, and Andreev.
NC i is the class of decision problems decidable by uniform boolean circuits with a polynomial number of gates of at most two inputs and depth O((log n) i), or the class of decision problems solvable in time O((log n) i) on a parallel computer with a polynomial number of processors. Clearly, we have